Closed-form linear stability conditions for rotating Rayleigh–Bénard convection with rigid stress-free upper and lower boundaries

Abstract

The linear dynamics of rotating Rayleigh–Bénard convection with rigid stress-free boundaries has been thoroughly investigated by Chandrasekhar (1961) who determined the marginal stability boundary and critical horizontal wavenumbers for the onset of convection and overstability as a function of the Taylor number $T$. No closed-form formulae appeared to exist and the results were tabulated numerically. However, by taking the Rayleigh number $R$ as independent variable we have found remarkably simple expressions. When the Prandtl number $P \geq P_{\!\!c}=0.67659$, the marginal stability boundary is described by the curve $T(R)= R[(R/R_c)^{1/2} -1]$ where $R_c=\frac{27}{4}\pi^4$ is Rayleigh's famous critical value for the onset of stationary convection in a non-rotating system ($T=0$). For $P < P_{\!\!c}$ the marginal stability boundary is determined by this curve until it is intersected by the curve \[ T(R,P)= R{\Bigg[\bigg(\frac{1+P}{2^3 P^4}\bigg)^{1/2}(R/R_c)^{1/2} -\frac{1+P}{2P^2}\Bigg]}. \]A simple expression for the intersection point is derived and also for the critical horizontal wavenumbers for which, along the marginal stability boundary, instability sets in either as stationary convection or in an oscillatory fashion. A simple formula is derived for the frequency of the oscillations. Further, we have analytically determined critical points on the marginal stability boundary above which an increase of either viscosity or diffusivity is destabilizing. Finally, we show that if the fluid has zero viscosity the system is always unstable, in contradiction to Chandrasekhar's conclusion.